This paper analyses low-frequency magnetohydrodynamic (MHD) modes, especially the geodesic acoustic modes (GAMs), in toroidal plasmas with large aspect ratio and circular cross section, including the effects of toroidal plasma rotation. A system of equations describing MHD modes with frequency of the order of the sound frequency in such plasmas is derived from the Frieman-Rotenberg equation, using a technique where the plasma perturbation ξ and the perturbed magnetic field Q are expanded separately in the inverse aspect ratio ε = r/R, where r and R denote the minor and major radii of the plasma torus, respectively. The large-scale, ideal MHD properties of the GAM induced by toroidal rotation (Wahlberg 2008 Phys. Rev. Lett. 101 115003) are thereafter analysed in more detail employing this system of equations. It is shown that both the axisymmetric GAMs existing in rotating plasmas are localized on a specific magnetic surface only to leading order in ε, and that a 'halo' consisting of finite components of both ξ and Q with dominant poloidal mode numbers m = ±2 appears outside this magnetic surface to higher orders in ε.

The stability of the internal m=n=1 kink mode is analyzed for a tokamak with a toroidally rotating plasma, by a large aspect ratio expansion of the compressible magnetohydrodynamic equations. Assuming that the central poloidal beta is of order unity, it is found that the internal kink mode is stabilized by rotational frequencies of order Ω/ωA∼ε, where ωA is the Alfvén frequency and ε is the inverse aspect ratio. The internal kink then turns into a stable oscillation with a Doppler-shifted frequency ∼ΩM(1−1/Γ)1/2, where Γ is the adiabatic index and ℳ is the sonic Mach number. The stabilization comes from the centrifugal force which gives a stable density (or entropy) distribution within each magnetic surface. The parallel motion associated with the internal kink mode then behaves as the Brunt–Väisälä oscillations of a stably stratified fluid in a gravitational field. At lower rotational frequencies, Ω/ωA∼ε2, the only effect of the rotation is a co-rotation of the usual (nonrotating) m=n=1 instability, whereas the ordering Ω/ωA∼ε3/2 represents a transition regime where the stabilizing effect of the rotation competes with the drive from the internal kink instability. Kinetic behavior along the field lines is expected to influence this stabilization mechanism, as it depends on the adiabatic index Γ.

The effect of toroidal rotation on the geodesic acoustic mode (GAM) in a tokamak is studied. It is shown that, in addition to a small frequency upshift of the ordinary GAM, another GAM, with much lower frequency, is induced by the rotation. The new GAM appears as a consequence of the nonuniform plasma density and pressure created by the centrifugal force on the magnetic surfaces. Both GAMs in a rotating plasma are shown to exist both as continuum modes with finite mode numbers m and n at the rational surfaces q=m/n as well as in the form of axisymmetric modes with m=n=0.

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A new dispersion relation, and associated stability criteria, is derived for low-n external kink and infernal modes, and is applied to modelling the stability properties of quiescent H-mode like regimes. The analysis, performed in toroidal geometry with large edge pressure gradients associated with a local flattening of the safety factor, includes a pedestal, sheared toroidal rotation and a vacuum region separating the plasma from an ideal metallic wall. The external kink-infernal modes found here exhibit similarities with experimentally observed Edge Harmonic Oscillations.